Let $K$ be a $d$-degree extension of $\mathbb{Q}_p$ with ring of integers $O_K$, unique maximal ideal $m_K$ and uniformizer $\pi$. Let $N=N_{K/\mathbb{Q}_p}$ be the field norm.
Consider the compact open set $U=1+\pi O_K=1+\pi \mathbb{Z}_p^d$, where $d=[K: \mathbb{Q}_p]$.
Question:
What would be the image $N(U)$ ?
For any $\alpha \in K$, the norm $N(\alpha)$ is the determinant of the multiplication by $\alpha$ map $x \mapsto \alpha x$.
If $K$ is Galois extension, then $$|N(\alpha)|=\prod_{\sigma \in \text{Gal}(K/\mathbb{Q}_p)}|\sigma(\alpha)|=|\alpha|^d,$$ where $|.|$ is the norm extending $|.|_p$.
Another simple case, if $U=1+p \mathbb{Z}_p$, then any $\sigma \in \text{Gal}(K/\mathbb{Q}_p)$ fix it i.e., $\sigma(1+p \mathbb{Z}_p)=1+p \sigma(\mathbb{Z}_p)=1+p \mathbb{Z}_p$ and therefore $$N(1+p \mathbb{Z}_p)=\prod_{\sigma \in \text{Gal}(K/\mathbb{Q}_p)}\sigma(1+p \mathbb{Z}_p)=\prod_{\sigma \in \text{Gal}(K/\mathbb{Q}_p)}(1+p \mathbb{Z}_p)=(1+p \mathbb{Z}_p)^d.$$ What is $(1+p \mathbb{Z}_p)^d$ ?
Edit: Serre's book on "Local Field" says $$N(1+\pi^n O_K) \subset 1+p^n \mathbb{Z}_p.$$ When is $N(1+\pi^n O_K) = 1+p^n \mathbb{Z}_p$ ?
But what would be the exact image $N(U)$, when $U=1+\pi O_K$ ?
I appreciate any comments. Thanks